Bimodal behavior of post-measured entropy and one-way quantum deficit for two-qubit X states

TL;DR

This paper develops a method to analyze the bimodal behavior of post-measured entropy in two-qubit X states, revealing phase transitions in the one-way quantum deficit and identifying subregions with variable optimal measurement angles.

Contribution

It introduces a new approach for calculating the one-way quantum deficit and uncovers bimodal entropy behavior leading to phase boundary identification in X states.

Findings

Bimodal entropy behavior with two interior extrema in certain X states.

Phase boundaries characterized by finite jumps in optimal measurement angles.

Subregions with variable optimal angles occupy about 1% of the state space, with high fidelity.

Abstract

A method for calculating the one-way quantum deficit is developed. It involves a careful study of post-measured entropy shapes. We discovered that in some regions of X-state space the post-measured entropy $\tilde S$ as a function of measurement angle $\theta\in[0,\pi/2]$ exhibits a bimodal behavior inside the open interval $(0,\pi/2)$, i.e., it has two interior extrema: one minimum and one maximum. Furthermore, cases are found when the interior minimum of such a bimodal function $\tilde S(\theta)$ is less than that one at the endpoint $\theta=0$ or $\pi/2$. This leads to the formation of a boundary between the phases of one-way quantum deficit via {\em finite} jumps of optimal measured angle from the endpoint to the interior minimum. Phase diagram is built up for a two-parameter family of X states. The subregions with variable optimal measured angle are around 1$\%$ of the total region, with their relative linear sizes achieving $17.5\%$, and the fidelity between the states of those subregions can be reduced to $F=0.968$. In addition, a correction to the one-way deficit due to the interior minimum can achieve $2.3\%$. Such conditions are favorable to detect the subregions with variable optimal measured angle of one-way quantum deficit in an experiment.

Tables

Table 1. Table 1: Jumps of optimal measured angles, Δ ​ ϑ Δ italic-ϑ \Delta\vartheta , on the boundary between the phases Δ 0 subscript Δ 0 \Delta_{0} and Δ ϑ subscript Δ italic-ϑ \Delta_{\vartheta}

$q_1$	$q_2$	$\Delta \vartheta$
$0.5$	$0$	$0=0^{\circ }$
$\mathrm{0.544\hspace{0.17em}535}$	$0.55-q_1$	$0.1267\approx 7^{\circ }$
$\mathrm{0.588\hspace{0.17em}104}$	$0.6-q_1$	$0.2470\approx {14}^{\circ }$
$\mathrm{0.631\hspace{0.17em}766}$	$0.65-q_1$	$0.4020\approx {23}^{\circ }$
$\mathrm{0.676\hspace{0.17em}082}$	$0.7-q_1$	$0.6252\approx {36}^{\circ }$
$\mathrm{0.721\hspace{0.17em}590}$	$0.75-q_1$	$1.0409\approx {60}^{\circ }$
$\mathrm{0.739\hspace{0.17em}409}$	$\mathrm{0.029\hspace{0.17em}686}$	$\pi /2={90}^{\circ }$

Equations

$$Q=\min \{Q_0,Q_{\vartheta },Q_{\pi /2}\}.$$

$$Q^{\prime \prime }(0)=0,Q^{\prime \prime }(\pi /2)=0.$$

$$\mathrm{\Delta }=\text{cases}\mathrm{\Delta }(\vartheta ),{\mathrm{\Delta }}^{\prime \prime }(0)<0\mathrm{and}{\mathrm{\Delta }}^{\prime \prime }(\pi /2)<0,\vartheta \in (0,\pi /2);\text{cr}\min \{\mathrm{\Delta }(0),\mathrm{\Delta }(\pi /2)\},\mathrm{others}.$$

$${\rho }_{AB}=q_1|{\Psi }^{+}⟩⟨{\Psi }^{+}|+q_2|{\Psi }^{-}⟩⟨{\Psi }^{-}|+(1-q_1-q_2)|00⟩⟨00|,$$

$$\rho_{AB}=\left(\begin{array}[]{cccc}1-q_{1}-q_{2}&0&0&0\\ 0&(q_{1}+q_{2})/2&(q_{1}-q_{2})/2&0\\ 0&(q_{1}-q_{2})/2&(q_{1}+q_{2})/2&0\\ 0&0&0&0\end{array}\right).$$

$${\lambda }_1=1-q_1-q_2,{\lambda }_2=q_1,{\lambda }_3=q_2,{\lambda }_4=0.$$

$$q_1\ge 0,q_2\ge 0,q_1+q_2\le 1.$$

$$\mathrm{\Delta }=\underset{\{{\mathrm{\Pi }}_{\mathrm{k}}\}}{\min }S({\tilde{\rho }}_{AB})-S({\rho }_{AB}),$$

$${\tilde{\rho }}_{AB}=\sum _k(I\otimes {\mathrm{\Pi }}_k){\rho }_{AB}(I\otimes {\mathrm{\Pi }}_k{)}^{+}$$

$${\mathrm{\Pi }}_k=V{\pi }_k{V}^{+},$$

$$V=\left(\begin{array}[]{cc}\cos(\theta/2)&-e^{-i\phi}\sin(\theta/2)\\ e^{i\phi}\sin(\theta/2)&\cos(\theta/2)\end{array}\right)$$

$$S(q_1,q_2)\equiv S({\rho }_{AB})=-q_1\log q_1-q_2\log q_2-(1-q_1-q_2)\log (1-q_1-q_2).$$

$${\Lambda }_{1,2}$$

$${\Lambda }_{3,4}$$

$$\tilde{S}(\theta ;q_1,q_2)\equiv S({\tilde{\rho }}_{AB})=h_4({\Lambda }_1,{\Lambda }_2,{\Lambda }_3,{\Lambda }_4),$$

$${\tilde{S}}^{\prime }(0)={\Delta }^{\prime }(0)\equiv 0,{\tilde{S}}^{\prime }(\pi /2)={\Delta }^{\prime }(\pi /2)\equiv 0.$$

$${\tilde{S}}_0(q_1,q_2)=-(1-q_1-q_2)\log (1-q_1-q_2)-(q_1+q_2)\log [(q_1+q_2)/2],$$

$${\tilde{S}}_{\pi /2}(q_1,q_2)=\log 2+h((1+\sqrt{(1-q_1-q_2{)}^2+(q_1-q_2{)}^2})/2),$$

$${\Delta }_0={\Delta }_{\pi /2}$$

$${\Delta }^{\prime \prime }(0)=0\mathrm{and}{\Delta }^{\prime \prime }(\pi /2)=0$$

$${\tilde{S}}^{\prime \prime }(\pi /2)$$

$$r=\sqrt{(1-q_1-q_2{)}^2+(q_1-q_2{)}^2}.$$

$${\tilde{S}}^{\prime \prime }(0)=\frac{1-3q+2q^2}{2-3q}\ln \frac{2(1-q)}{q},$$

$${\Delta }_0={\Delta }_{\vartheta }.$$

$$\mathrm{\Delta }=\min \{{\Delta }_0,{\Delta }_{\vartheta },{\Delta }_{\pi /2}\},$$

$${\rho }_{AB}=q_1|{\Psi }^{+}⟩⟨{\Psi }^{+}|+q_2|{\Psi }^{-}⟩⟨{\Psi }^{-}|+q_3|00⟩⟨00|+q_4|11⟩⟨11|$$

Full text

1 Introduction

Quantum correlation is a key feature of quantum mechanics and it lies at the heart of quantum information science. Besides the quantum entanglement and discord, the one-way quantum deficit is one of the most important measures of quantum correlation MBCPV12 ; Str15 ; ABC16 ; BDSRSS17 . The entanglement is identical to the discord and one-way deficit for the pure quantum states, whereas the discord and one-way deficit coincide in considerably more general cases — they are the same for the Bell-diagonal states and even for the X states with zero Bloch vector for one qubit (i.e., with a single maximally mixed marginal) if the local measurements are performed on this qubit YF16 .

Definitions of quantum discord $Q$ and one-way quantum deficit $\rm\Delta$ involve the minimization procedure to obtain the optimal measurement performed on one part of bipartite system. This procedure for the two-qubit systems with X density matrix is reduced to the minimization problem on one variable – the polar angle $\theta\in[0,\pi/2]$ (see Refs. CRC10 ; VR12 ; JY16 ; WJFWCF15 ). Moreover, a formula for the quantum discord is presented in a partially analytic (piecewise-analytical-numerical) form Y14 ; Y14a ; Y15 ,

[TABLE]

Here, the subfunctions (branches) $Q_{0}$ and $Q_{\pi/2}$ are the analytical expressions (corresponding to the discord with optimal measurement angles equaling zero and $\pi/2$, respectively) and only the third branch $Q_{\vartheta}$ requires to perform numerical minimization to obtain state-dependent minimizing angle $\vartheta\in(0,\pi/2)$ if, of course, the interior minimum exists. Equations for 0- and $\pi/2$-boundaries separating respectively the $Q_{0}$ and $Q_{\pi/2}$ regions with the $Q_{\vartheta}$ one can be written as Y14 ; Y14a ; Y15

[TABLE]

Here $Q^{\prime\prime}(0)$ and $Q^{\prime\prime}(\pi/2)$ are the second derivatives of the measurement-dependent discord function $Q(\theta)$ with respect to $\theta$ at the endpoints $\theta=0$ and $\pi/2$, correspondingly. The equations (2) are based on the unimodality hypothesis for the function $Q(\theta)$ which is confirmed for different classes of X states Y15 ; Y17 . Notice that Eqs. (2) reflect the bifurcation mechanism of appearance of the minimum inside the interval $(0,\pi/2)$.

On the other hand, as mentioned above, there is a close connection between the one-way quantum deficit and quantum discord. Therefore it would be tempting to propose that similar properties are valid for the measurement-dependent one-way quantum deficit function $\Delta(\theta)={\tilde{S}}(\theta)-S$, where $S$ is the pre-measurement entropy.

Recently, the authors YWF16 have claimed the result which is reduced to the statement that the one-way quantum deficit ${\rm\Delta}=\min_{\theta}\Delta(\theta)$ for the general X states is given by

[TABLE]

If the function $\Delta(\theta)$ is monotonic or has single extremum inside the interval $(0,\pi/2)$ this conclusion takes place.

In the present paper we show that the post-measured entropy and consequently the measurement-dependent one-way quantum deficit can display more general behavior which refutes the relation (3). We discuss the difficulties arisen from a new type of behavior and propose, instead of Eq. (3), the method giving the correct calculation of one-way deficit for two-qubit X states.

2 Results and discussion

Let us consider a two-parameter family of X states

[TABLE]

where $|\Psi^{\pm}\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2}$. This family generalizes the class of special X states from Ref. YWF16 which corresponds to the case $q_{1}=0$.

The density matrix (4) in open form is given as

[TABLE]

Eigenvalues of this matrix equal

[TABLE]

Owing to the non-negativity requirement for any density matrix, one obtains that the domain of definition for the parameters (arguments) $q_{1}$ and $q_{2}$ is restricted by conditions

[TABLE]

Thus, the domain in plane $(q_{1},q_{2})$ is the triangle ${\cal T}$ which is shown in Fig. 1.

One-way quantum deficit (quantum work deficit) for a bipartite state $\rho_{AB}$ is defined as the minimal increase of entropy after a von Neumann measurement on one party (without loss of generality, say, $B$) SKB11 ; CPBSS15 ; CDSPSS15

[TABLE]

where

[TABLE]

is the weighted average of post-measured states and $S(\cdot)$ means the von Neumann entropy. In Eqs. (8) and (9), $\rm\Pi_{k}$ ($k=0,1$) are the general orthogonal projectors

[TABLE]

where $\pi_{k}=|k\rangle\langle k|$ and transformations $\{V\}$ belong to the special unitary group $SU_{2}$. Rotations $V$ may by parametrized by two angles $\theta$ and $\phi$ (polar and azimuthal, respectively):

[TABLE]

with $0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$.

Using Eq. (6) one gets the pre-measured entropy

[TABLE]

Eigenvalues of the matrix ${\tilde{\rho}}_{AB}$ are equal to

[TABLE]

It is seen that the azimuthal angle $\phi$ has dropped out from the given expressions. This is due to the fact that one pair of non-diagonal entries of the density matrix (5) vanishes. Using Eqs. (2) we arrive at the post-measured entropy (entropy after measurement)

[TABLE]

where $h_{4}(x_{1},x_{2},x_{3},x_{4})=-\sum_{i=1}^{4}x_{i}\log x_{i}$ with additional condition $x_{1}+x_{2}+x_{3}+x_{4}=1$ is the quaternary entropy function.

Notice that function $\tilde{S}$ of argument $\theta$ is invariant under the transformation $\theta\to\pi-\theta$ therefore it is enough to restrict oneself by values of $\theta\in[0,\pi/2]$. Moreover, the pre- and post-measured entropies $S$ and $\tilde{S}$, as functions of $q_{1}$ and $q_{2}$, are symmetric under the exchange $q_{1}\rightleftharpoons q_{2}$.

Equations (12)–(14) define the measurement-dependent one-way deficit function $\Delta(\theta)={\tilde{S}}(\theta)-S$. Direct calculations show that for every choice of model parameters the function ${\tilde{S}}(\theta)$ and hence $\Delta(\theta)$ possess an important property, namely their first derivatives with respect to $\theta$ identically equal zero at both endpoints $\theta=0$ and $\theta=\pi/2$:

[TABLE]

From Eqs. (2) and (14) we get the expressions for the post-measurement entropy at the endpoint $\theta=0$,

[TABLE]

and at the second endpoint $\theta=\pi/2$:

[TABLE]

where $h(x)=-x\log x-(1-x)\log(1-x)$ is the Shannon binary entropy function. Together with Eq. (12) they supply us with explicit expressions for the one-way deficit at the endpoints: $\Delta_{0}=\Delta(0)$ and $\Delta_{\pi/2}=\Delta(\pi/2)$. In particular, if $q_{1}$ or $q_{2}$ equals zero then $\Delta_{0}=q\log 2~{}(=q,{\rm bit})$, where $q=\{q_{1},q_{2}\}$.

Solving the transcendental equation

[TABLE]

or, the same, $\tilde{S}_{0}=\tilde{S}_{\pi/2}$ we find the subregions in the plane $(q_{1},q_{2})$, where $\Delta_{\pi/2}<\Delta_{0}$ (restricted in Fig. 1 by dotted curves 1 and 1*′* and corresponding Cartesian axes $Oq_{1}$ and $Oq_{2}$) and where, v.v., $\Delta_{0}<\Delta_{\pi/2}$ (marked in Fig. 1 by symbol $\Delta_{0}$). The curve 1 has two endpoints on the axis $Oq_{1}$: at $q_{1}=0.61554$ and $q_{1}=1$. Analogously for the curve 1*′* (see Fig. 1).

The 0- and $\pi/2$-boundaries, i.e., where respectively the second derivatives

[TABLE]

or, the same, $\tilde{S}^{\prime\prime}(0)=0$ and $\tilde{S}^{\prime\prime}(\pi/2)=0$, will be needed below. As calculations yield,

[TABLE]

where

[TABLE]

On the other hand, calculations show that the second derivative $\tilde{S}^{\prime\prime}(\theta)$ with respect to $\theta$ is finite at $\theta=0$ only when $q_{1}q_{2}=0$:

[TABLE]

where again $q=\{q_{1},q_{2}\}$. The roots of equation $\tilde{S}^{\prime\prime}(0)=0$ are 1/2 and 1. Thus, the bifurcation 0-boundary exists only if $q_{1}=0$ or, inversely, $q_{2}=0$ (that is, only at two points on each of the Cartesian axes $Oq_{1}$ and $Oq_{2}$). The corresponding 0-boundaries $q_{1}=1/2$, when $q_{2}=0$, and $q_{2}=1/2$, when $q_{1}=0$ are shown in Fig. 1 by the crosses.

The results of numerical solution of the equation $\tilde{S}^{\prime\prime}(\pi/2)=0$ are presented in Fig. 1 by solid lines 2 and 2*′. The endpoints for the curve 2 on the axis $Oq_{1}$ are $q_{1}=0.67515$ and $q_{1}=1$. The curves 1 and 2 intersect at the point with coordinates $q_{1}=0.739\,409$ and $q_{2}=0.029\,686$ ($q_{1}+q_{2}=0.769\,095$). Analogously for the curves 1′* and 2*′* with, of course, permutation of $q_{1}$ and $q_{2}$ (see again Fig. 1).

Let us consider the behavior of post-measured entropy $\tilde{S}(\theta)$ and non-minimized one-way deficit $\Delta(\theta)$ by moving along different trajectories (paths) in the triangle ${\cal T}$.

Start with the passing along the leg of triangle ${\cal T}$. Figure 2 shows the evolution of shape of the post-measured entropy $\tilde{S}(\theta;q_{1},0)$ with changing the parameter $q_{1}$. The curve has the monotonically increasing behavior when the argument $q_{1}$ varies from $q_{1}=0$ to $q_{1}=1/2$; see Fig. 2(a). At the point $q_{1}=1/2$ a bifurcation of the minimum at $\theta=0$ occurs. Then, when $q_{1}$ increases from 0.5 to 0.67515, the curve $\tilde{S}(\theta)$ has, as shown in Figs. 2(b) and (c), the interior minimum, with the function $\tilde{S}(\theta)$ being here unimodal. So, the region with variable optimal angle $\vartheta$ takes up a part $0.17515\approx 17.5\%$ on the section $[0,1]$ of $Oq_{1}$ axis and the fidelity of states at points $(0.5,0)$ and $(0.67515,0)$ is equal to $F=96.8\%$111 Note for comparison that in two-photon experiments one achieves now the values of fidelity $F=99.8(2)\%$ BSPBG13 and $F=99.8(1)\%$ Guo16 . . The position of such a local minimum smoothly increases from zero to $\pi/2$; see again the curves in Figs. 2(b) and (c). The values of $\tilde{S}_{0}$ and $\tilde{S}_{\pi/2}$ become equal at the point $q_{1}=0.61554$ ($\tilde{S}_{0}=\tilde{S}_{\pi/2}=1.57667$ bit, hence $\Delta_{\pi/2}=\Delta_{0}=q_{1}=0.61554$ bit) and the depth of interior minimum is 0.01397 bit what gives a relative correction to the one-way deficit equaled $\delta{\rm\Delta}=2.3\%$. Then, at the value of $q_{1}=0.67515$, the system experiences a new sudden transition – from the branch, which is characterized by the continuously changing optimal angle $\vartheta$ in the full interval (from 0 to $\pi/2$), to the branch $\tilde{S}_{\pi/2}$ with constant optimal measurement angle equaled $\pi/2$. After this the curves of post-measured entropy exhibit monotonically decreasing behavior as illustrated in Fig. 2(d). One should emphasize here that the minimized one-way quantum deficit, ${\rm\Delta}=\min_{\theta}\Delta(\theta)$, vs the model parameter $q_{1}$ is continuous and smooth. Nevertheless, the function ${\rm\Delta}(q_{1})$ has nonanalyticities at the points $q=0.5$ and 0.67515 which manifest themselves in higher derivatives.

Consider now the behavior of post-measurement entropy and measurement-dependent one-way deficit in the bulk area of ${\cal T}$. We can inspect the total domain taking all possible straight-line trajectories $q_{1}+q_{2}=const\leq 1$. The behavior of the system is, obviously, symmetric relative to the middle of such trajectories. Take, for instance, the trajectory $q_{1}+q_{2}=0.75$ which is shown in Fig. 1 by the straight line 3. The shape of the curve $\Delta(\theta)$ has the monotonically increasing type in the middle of this trajectory ($q_{1}=q_{2}=0.375$). However, with the increase of the value of parameter $q_{1}$, the birth of a pair of extrema from an inflection point occurs inside the interval $(0,\pi/2)$; the situation is illustrated in Fig. 3. This phenomenon happens at the value of $q_{1}=0.72015$. According to the definition (see, e.g., Ref. V86 ) a function having two extrema in some interval is called bimodal on this interval.

With further increase of the $q_{1}$ value a qualitatively new effect is observed. We demonstrate it by the curves $\tilde{S}(\theta)$ shown in Fig. 4.

When the parameter $q_{1}$ achieves the value of 0.72159, the position of global minimum suddenly jumps through a finite step $\Delta\vartheta$ from zero to $\vartheta=1.0409\approx 60^{\circ}$ (see Fig. 4). As a result, the fracture is arisen on the continuous curve of minimized one-way quantum deficit ${\rm\Delta}(q_{1})$. The position of the fracture point is determined from the equation $\tilde{S}_{0}=\tilde{S}_{\vartheta}$ or

[TABLE]

After this the interior minimum lies lower than another minimum located at the endpoint $\theta=0$. Notice that behavior of curve 3 in Fig. 4 leads to a contradiction with Eq. (3), i.e., the equation is incorrect for general X states.

With further increasing $q_{1}$ the interior minimum smoothly moves to the point $\theta=\pi/2$ and disappears at $q_{1}=0.72358$ when the trajectory crosses the curve 2, i.e., the $\pi/2$-boundary (see Fig. 1). The dynamics of corresponding deformations of $\Delta(\theta)$ is depicted in Fig. 5. After crossing the $\pi/2$-boundary, the behavior of $\rm\Delta$ undergoes to the branch $\Delta_{\pi/2}$ up to the point of contact of trajectory with the Cartesian axis, i.e., up to $q_{1}=0.75$, where the interior maximum of $\Delta(\theta)$ disappears at the endpoint $\theta=0$. This happens through a new non-bifurcation (and non-inflection) mechanism. Since the second derivative $\Delta^{\prime\prime}(\theta)$ at $\theta=0$ diverges out of the Cartesian axes we will call this mechanism the singular one.

As a result, the one-way quantum deficit is obtained from the final equation

[TABLE]

where $\Delta_{0}$ and $\Delta_{\pi/2}$ are known in closed analytical forms and $\Delta_{\vartheta}$ is found numerically. The behavior of one-way deficit along the trajectory $q_{1}+q_{2}=0.75$ is shown in Fig. 6.

Either totally or partially similar behavior takes place for other trajectories $q_{1}+q_{2}=const$ which go lower the intersection point of curves defined by equations $\Delta_{0}=\Delta_{\pi/2}$ and $\Delta^{\prime\prime}(\pi/2)=0$, i.e, when $const\leq 0.769\,095$. For example, in the case of trajectory $q_{1}+q_{2}=0.65$, the bimodality appears at $q_{1}\simeq 0.631$ and a jump of optimal measurement angle from zero happens at $q_{1}=0.631\,766$. Values of jump angles $\Delta\vartheta$ in different cases are collected in Table 1.

A set of points where the optimal measurement angle discontinuously changes from zero to a finite value gives the jumping (or hopping) boundary; it serves instead of the absent ordinary 0-boundary (see Fig. 7).

Between this boundary and the $\pi/2$-one, there exists an intermediate phase (fraction) $\Delta_{\vartheta}$ with state-dependent optimal measurement angle $\vartheta$ which smoothly varies from some nonzero value to $\pi/2$. The flat of two subregions with variable optimal angle, $\Delta_{\vartheta}$, is near $1\%$ of the one of total domain ${\cal T}$.

When $const>0.769\,095$ (i.e., when the trajectories lie above the black circle shown in Fig. 7), the situation is different. With increasing $q_{1}$ from middle values to the endpoint on the axis $Oq_{1}$ the curves $\tilde{S}(\theta)$ or $\Delta(\theta)$ are deformed from monotonically increasing shape to the shape with a single interior maximum (which is born at the point, where $\Delta^{\prime\prime}(\pi/2)=0$) and then a sudden transition $\Delta_{0}\to\Delta_{\pi/2}$ occurs at the boundary defined by the relation $\Delta_{0}=\Delta_{\pi/2}$ (line 3 in Fig. 7). Here, there is no intermediate region $\Delta_{\vartheta}$ and the transition $\Delta_{0}\to\Delta_{\pi/2}$ is characterized visually by a fracture on the curve ${\rm\Delta}(q_{1})$. Typical behavior of one-way deficit is shown in Fig. 8 along the trajectory $q_{1}+q_{2}=0.8$.

So, the presented method to calculate the one-way quantum deficit of X states is reduced first of all to careful analyzing of the shapes of post-measured entropy or measurement-dependent one-way deficit curves. One should also solve equations for the boundaries between three possible phases (branches): Eqs. (18), (19), and (23). After this the one-way quantum deficit is obtained from the piecewise-analytical-numerical formula (24).

3 Summary and concluding remarks

In this paper we have found that besides the monotonic and unimodal behavior the post-measured entropy and hence the measurement-dependent one-way quantum deficit upon the measurement angle can have a new kind of behavior. Namely, these functions can exhibit the bimodal shape in the open interval $(0,\pi/2)$ for different regions in the space of X state parameters. This expands the variety of behavior for the one-way quantum deficit $\rm\Delta$. In particular, a new state-dependent phase (fraction) which is characterized by a partial interval of optimal measured angles has been found. Instead of smooth conjugation of the branches $\Delta_{0}$ and $\Delta_{\pi/2}$ this leads to a fracture on the curve of one-way deficit.

New mechanism of a boundary arising between the phases via jumping the optimal measured angle on a finite step has been discovered. Instead of bifurcation conditions (19) the boundary is now determined by a relation like (23). The study of post-measured entropy shapes is the general way to determine the correct one-way quantum deficit.

This is in contrast with the behavior of conditional entropy and, consequently, measurement-dependent quantum discord in the same regions of parameter space: their behavior is restricted by monotonic and unimodal types. In any case, this rather simple and therefore attractive picture is valid for the different specific cases and subclasses of X states Y14 ; Y15 ; Y17 . In particular, such a behavior of conditional entropy is confirmed for the symmetric XXZ states Y17 those may be written in an equivalent form as

[TABLE]

with $q_{1}+q_{2}+q_{3}+q_{4}=1$.

An intriguing question remains: are there any more general shapes of curves for the post-measured entropy of X states? For instance, can this entropy have trimodal and, maybe, multimodal dependence? The answer to these and other questions should come from the future investigations of post-measurement entropy shapes in the full five-parameter X-state space.

Acknowledgment The work was supported by the Russian Foundation for Basic Research.